Applied Symbolic Dynamics and Chaos
Symbolic dynamics is a coarse-grained description of dynamics. It provides a rigorous way to understand the global systematics of periodic and chaotic motion in a system. In the last decade it has been applied to nonlinear systems described by one- and two-dimensional maps as well as by ordinary differential equations. This book will help practitioners in nonlinear science and engineering to master that powerful tool.
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Symbolic Dynamics of Unimodal Maps
Maps with Multiple Critical Points
Symbolic Dynamics of Circle Maps
Symbolic Dynamics of TwoDimensional Maps
Application to Ordinary Differential Equations
Counting the Number of Periodic Orbits
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admissibility conditions backward foliations backward sequences backward symbolic sequence bifurcation diagram chaos chaotic attractor circle map compatible kneading pair composition rule construct corresponding cubic map defined denote determined dynamical foliations eventually periodic example Figure fixed point follows forbidden zone forward foliations fundamental forbidden zone gap map given Henon map infinite initial point intersection interval invariant iterations kneading plane Lorenz equations Lorenz model Lozi map manifolds mapping function maps with multiple matrix maximal median words metric representation monotone branches multiple critical points nonlinear number of periods one-dimensional maps orbital points ordering rule parity partition line period-doubling bifurcation periodic orbits periodic sequences Periodic Window phase plane phase space piecewise linear quadratic map real numbers return map rotation number shift-maximality shown in Fig stable subinterval superstable kneading sequences superstable periodic surjective symbolic dynamics symbolic plane symmetry breaking tangencies tent map topological entropy unimodal map unstable periodic orbits well-ordered